The MSA based on Pontryagin's maximum principle [1]} for seeking numerical solutions to deterministic control systems was first proposed by Krylov et al. [2]}. After that, many improved modifications of the MSA have been developed by researchers for a variety of deterministic control systems ([3]}, [4]}, [5]}, [6]}).
The key idea is to find a set of small measure where the control is varied finitely so as to reduce the value of the functional in each iteration.
Recently, a modified MSA was first extended to solving the stochastic control problems with convex control domains [7]}. Their convergence result shows, roughly speaking, the controls \(\left\lbrace  u^{m}(\cdot ) \right\rbrace _{m \in \mathbb {N}}\)  produced by the MSA algorithm increasingly satisfy the extended stochastic maximum principle as one moves from one iteration to the next. Yet, as Theorem 2.5 in [7]} stated, the algorithm converges to a local minimum of the concerned cost functional \(J\) . Motivated by this, not only are we aimed at establishing the MSA for the stochastic control problems with general control domains, but also giving a sufficient condition from which, at least, the sequence \(J(u^{m}(\cdot ))\)  is bound to descend to the global minimum \(J(\bar{u}(\cdot ))\)  rather than any local minimum, where \(\bar{u}(\cdot )\)  is the optimal control.
To be more clear, for any \(\delta >0\) , we prove that there exists a positive integer \(N_{\delta }\)  such that,
if \(m \ge N_{\delta }\) , then \(u^{m}(\cdot ) \)  minimizes the \(\mathcal {H}\) -function in terms of
\(\begin{array}[c]{l}\mathbb {E}\left[ \int _{0}^{T}\mathcal {H}(t,X^{m}(t),p^{m}(t),q^{m}(t),P^{m}(t),u^{m}(t),u^{m}(t))dt\right] \\\le \inf \limits _{u(\cdot )}\mathbb {E}\left[ \int _{0}^{T}\mathcal {H}(t,X^{m}(t),p^{m}(t),q^{m}(t),P^{m}(t),u(t),u^{m}(t))dt\right]+\delta ,\end{array}\) 
